I see what you're saying now. So to answer your original question...
> You don't think it's easier to say...
No. I disagree, your way of thinking is harder for me to think. :-)
You're correct, but my mind didn't work like yours. But that's the beautiful thing about mathematics: we both are correct. We just had different viewpoints about how things work. Ultimately, it seems like we're both saying the same thing, although we tweaked the formulas to look like the simplest ways for our own brains.
I feel like I'm not understanding something. Isn't "n·0 = 0" the demonstrandum? I would have written something like
n·0 = 0 [we seek to show this]
-------
n·0 = n·(1 + -1) [definition of additive inverse]
n·(1 + -1) = n + (-n) [multiplication is distributive...]
n + (-n) = 0 [definition of additive inverse]
QED
(Sorry, but what you have above makes me slightly queasy, and I'm hoping the fuss about rigor can be acceptable in a math thread.)
I see. I'm not a real mathematician, I just dabble in it on occasion. And I fully admit: my math professors always criticized my proof constructions throughout my life, so its definitely not something I was ever that good at.
I'll happily take your advice however! Proper mathematical rigor is always something I appreciate, even if its something I'm not very much practiced at.
> You don't think it's easier to say...
No. I disagree, your way of thinking is harder for me to think. :-)
You're correct, but my mind didn't work like yours. But that's the beautiful thing about mathematics: we both are correct. We just had different viewpoints about how things work. Ultimately, it seems like we're both saying the same thing, although we tweaked the formulas to look like the simplest ways for our own brains.